When I save something as an OBJ file the texture does not save with it, why?
.OBJ (Wavefront) file itself only stores geometry data, it does not keep anything regarding materials or textures.

Materials (including references [paths] to textures) are stored in companion .MTL file/format. But it does not embed textures themselves either.

So you have to give .obj, .mtl and all your texture files (you can use 'Copy' option of the exporter to copy all needed textures in a subdir next to exported .obj).

But there is yet another issue with OBJ - filenames must absolutely not contain any space. This is simply not supported by the format, and if your mtl and/or texture files have some, they won't get loaded.